Abstract

We investigate the nonlocal boundary value problems of impulsive fractional differential equations. By Banach’s contraction mapping principle, Schaefer’s fixed point theorem, and the nonlinear alternative of Leray-Schauder type, some related new existence results are established via a new special hybrid singular type Gronwall inequality. At last, some examples are also given to illustrate the results.

1. Introduction

Fractional differential equations have recently proved to be strong tools in the modeling of many physical phenomena. It draws a great application in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow in porous media, and fluid dynamic traffic model. For more details on fractional calculus theory, one can see the monographs of Diethelm [1], Kilbas et al. [2], Lakshmikantham et al. [3], Miller and Ross [4], Podlubny [5], and Tarasov [6]. Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attentions (see, e.g., [7–13]).

The impulsive differential equations arise from the real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are naturally seen in biology, physics, engineering, and so forth. Due to their significance, many authors have established the solvability of impulsive differential equations. For the general theory and applications of such equations we refer the interested readers to see the papers [14–17] and references therein.

As one of the important topics in the research of differential equations, the boundary value problems have attained a great deal of attention from many researchers; see [18–23] and the references therein. As pointed out in [24], the nonlocal boundary condition can be more useful than the standard condition to describe some physical phenomena. But there are very few papers (see, e.g., [24–26]) dealing with the nonlocal boundary value problems of fractional differential equations. And even in [24–26], the impulsive effect has not been considered. In [27], the author considered the following problems: where , is a continuous function, and are continuous functions, , , , , , , , and are two continuous functions, , , ,  and exist with , .

In [27], by a fixed point theorem due to O’Regan, the authors established sufficient conditions for the existence of at least one solution for the problem (1).

In [28], the authors considered the following problem: where is the Caputo fractional derivative of order with the lower limit zero, is jointly continuous, satisfy ,   and represent the right and left limits of at , , and , , are real constants with .

In [29], the authors studied the following problem: where is the Caputo fractional derivative of order with the lower limit zero, , is jointly continuous, , and satisfy , and with , representing the right and left limits of at . In [29], the authors obtained the sufficient condition of the existence of at least one solution for problem (3).

Motivated by the work mentioned above, we consider the following impulsive fractional differential equation with nonlocal boundary value conditions: where , is a continuous function, and are continuous functions; and with , , , , , , and , and are two continuous functions, ; , , and , exist with , . The main methods in our paper are Banach’s contraction mapping principle, Schaefer’s fixed point theorem, and the nonlinear alternative of Leray-Schauder type.

Obviously, the problems in our paper are different from those in [27], and we generalized the methods and results in [27]. Problems in our paper are more universal than problems in [28, 29]. It should also be noted that the basic space in our paper is , , , exist, and is left continuous at ,  , which is a Banach space with the norm , where , . The basic space in [29] is , , and exist with , , with the norm , which is unreasonable for the order because , may not exist, for . So the problem (3) in [29] are not well defined, and Definition 4.1 is also unreasonable and should be modified.

The rest of this paper is organized as follows. In Section 2, we will give some lemmas which are essential to prove our main results. In Section 3, we give the main results. The first result is based on the Banach contraction principle, the second result is based on Schaefer’s fixed point theorem via a generalized hybrid singular Gronwall inequality, and the third result is based on a nonlinear alternative of Leray-Schauder type. In Section 4, some examples are offered to demonstrate the application of our main results.

2. Preliminaries

At first, we present the necessary definitions for the fractional calculus theory.

Definition 1 (see [2, 5]). The Riemann-Liouville fractional integral of order of a function is given by where the right side is pointwise defined on .

Definition 2 (see [2, 5]). The Caputo fractional derivative of order of a function is given by where ,   denotes the integer part of number , and the right side is pointwise defined on .

Lemma 3 (see [2, 5]). Let ; then the fractional differential equation has solutions where , , and .

Lemma 4 (see [2, 5]). Let . Then one has where , , and .

Lemma 5 (see [29, Lemma 2.9]). Let satisfy the following inequality: where , ; for some , are constants. Then there exists a constant such that

Lemma 6 (Schaefer’s fixed point theorem). Let be a Banach space and let be a completely continuous operator. If the set is bounded, then has at least a fixed point.

Lemma 7 (nonlinear alternative of Leray-Schauder type). Let be a nonempty convex subset of . Let be a nonempty open subset of with and let be a compact and continuous operator. Then either(i)has fixed points, or(ii)there exist and with .
We define , , and exist with . ; , , , exist and is left continuous at , .
Obviously, is a Banach space with the norm , where , .

Then we can define the solution for the problem (4).

Definition 8. A function with its Caputo derivative of order existing on is a solution of the problem (4) if for and satisfies a.e. on , the restriction of on is just , and the conditions , ,  , with , .

Lemma 9. For any , a function is a solution of the nonlocal impulsive problem if and only if is a solution of the fractional integral equation with where

Proof. By Lemmas 3 and 4, the solution of (12) can be written as where . Taking the derivative of gives If , then we have where . In view of the impulse conditions and (16)–(18), we have Taking (20) into (18), we can get Repeating the process in this way, the solution for can be written as By taking the derivative of (22), we have Taking (16), (17), (22), and (23) to the boundary value conditions we can get Then the solution of (12) is (22), where , are given by (25). Taking derivative of (13), we can get
Conversely, taking (13) and (14) into (12), we can easily get the equation and all the impulse conditions and boundary value conditions are satisfied. So we complete the proof of Lemma 9.